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Expanding Brackets

Expanding means multiplying out brackets so an expression has no brackets left. It is the reverse of factorising and shows up everywhere — from solving equations to differentiating. The golden rule never changes: every term inside must be multiplied by every term outside.

30 min Video by Zeeshan Zamurred Algebraic Expressions

Edexcel AS Level Maths: 1.2 Expanding Brackets (Double/Triple)Watch the full walkthrough before the notes below.

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What you'll be able to do

Expand a single bracket by multiplying through
Expand two brackets using a reliable method (FOIL / grid)
Expand and simplify expressions with three brackets
Collect like terms accurately after expanding
Handle negative signs and coefficients without slips

Expanding a single bracket

To expand a single bracket, multiply the term outside by $each$ term inside. Watch the signs carefully — a negative outside the bracket flips the sign of everything inside.

a (b + c) = ab + a c

The multiplier

a

is distributed across every term in the bracket.

1Multiply

3 x

2 x

6 x^{2}

2Multiply

3 x

- 5

- 15 x

Answer

6 x^{2} - 15 x

Tip — A negative sign outside the bracket is a multiplier of −1. Treat it like any other factor and the signs take care of themselves.

Expanding two brackets

For two brackets, every term in the first must multiply every term in the second — four products in total. A common memory aid is $FOIL$ : First, Outer, Inner, Last. A grid (box) method does exactly the same job and is harder to rush.

(x + a) (x + b) = x^{2} + (a + b) x + ab

The middle term is the sum

a + b

; the constant is the product

ab

1First:

x \times x = x^{2}

2Outer + Inner:

7 x + 4 x = 11 x

3Last:

4 \times 7 = 28

Answer

x^{2} + 11 x + 28

Tip — Always combine the two “middle” terms. Leaving them separate is the single most common expanding error.

A useful special case

Squaring a bracket is just expanding it by itself. The shortcut $(x + a)^{2} = x^{2} + 2 a x + a^{2}$ saves time, but only if you remember the middle $2 a x$ term.

The “difference of two squares” pattern is worth memorising too: the middle terms cancel.

(x + a)^{2} = x^{2} + 2 a x + a^{2}

Square the bracket — do not forget the middle term.

(x + a) (x - a) = x^{2} - a^{2}

Difference of two squares — the

x

terms cancel.

(x + 5)^{2} = (x + 5) (x + 5)

2Middle term:

2 \times 5 \times x = 10 x

3Last:

5^{2} = 25

Answer

x^{2} + 10 x + 25

Tip — $(x + 5)^{2}$ is NOT $x^{2} + 25$ . The cross terms give the $10 x$ in the middle.

Expanding three brackets

With three brackets, expand $two$ of them first, then multiply the result by the third bracket. Keep your work neat and collect like terms only at the very end.

1Expand the first two:

(x + 2) (x - 1) = x^{2} + x - 2

2Now multiply by

(x + 3)

, term by term.

x^{2} (x + 3) = x^{3} + 3 x^{2}

;

x (x + 3) = x^{2} + 3 x

;

- 2 (x + 3) = - 2 x - 6

4Collect like terms:

x^{3} + (3 + 1) x^{2} + (3 - 2) x - 6

Answer

x^{3} + 4 x^{2} + x - 6

Tip — It does not matter which two brackets you expand first — pick the pair that looks easiest.

Formula recap

a (b + c) = ab + a c

Single bracket — multiply through.

(x + a) (x + b) = x^{2} + (a + b) x + ab

Two brackets — FOIL / grid.

(x + a)^{2} = x^{2} + 2 a x + a^{2}

Squaring a bracket.

(x + a) (x - a) = x^{2} - a^{2}

Difference of two squares.

Common mistakes to avoid

Only multiplying the first term inside a bracket, e.g. 3(x + 2) = 3x + 2.

Multiply every term: 3(x + 2) = 3x + 6.

Writing (x + 5)² = x² + 25.

Expand fully: (x + 5)² = x² + 10x + 25.

Mishandling a negative outside a bracket: −(x − 4) = −x − 4.

Flip every sign inside: −(x − 4) = −x + 4.

Key takeaways

Every term inside the bracket is multiplied by every term outside.
Two brackets give four products — remember to combine the two middle terms.
Squaring a bracket keeps a middle term; difference of two squares loses it.
For three brackets, expand a pair first, then multiply by the third.

Test yourself

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